Perotto Simona, Zilio Alessandro
MOX, Department of Mathematics, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milan, Italy.
Centre d'Analyse et de Mathématique Sociales, École des Hautes Études en Sciences Sociales, 190-198 Avenue de France, 75244 Paris Cedex 13, France.
Adv Model Simul Eng Sci. 2015;2:25. doi: 10.1186/s40323-015-0046-4. Epub 2015 Oct 13.
Surrogate solutions and surrogate models for complex problems in many fields of science and engineering represent an important recent research line towards the construction of the best trade-off between modeling reliability and computational efficiency. Among surrogate models, hierarchical model (HiMod) reduction provides an effective approach for phenomena characterized by a dominant direction in their dynamics. HiMod approach obtains 1D models naturally enhanced by the inclusion of the effect of the transverse dynamics.
HiMod reduction couples a finite element approximation along the mainstream with a locally tunable modal representation of the transverse dynamics. In particular, we focus on the pointwise HiMod reduction strategy, where the modal tuning is performed on each finite element node. We formalize the pointwise HiMod approach in an unsteady setting, by resorting to a model discontinuous in time, continuous and hierarchically reduced in space (c[M([Formula: see text])G()]-dG() approximation). The selection of the modal distribution and of the space-time discretization is automatically performed via an adaptive procedure based on an analysis of the global error. The final outcome of this procedure is a table, named , that sets the time partition and, for each time interval, the corresponding 1D finite element mesh together with the associated modal distribution.
The results of the numerical verification confirm the robustness of the proposed adaptive procedure in terms of accuracy, sensitivity with respect to the goal quantity and the boundary conditions, and the computational saving. Finally, the validation results in the groundwater experimental setting are promising.
The extension of the HiMod reduction to an unsteady framework represents a crucial step with a view to practical engineering applications. Moreover, the results of the validation phase confirm that HiMod approximation is a viable approach.
在许多科学和工程领域，针对复杂问题的替代解决方案和替代模型是近期一项重要的研究方向，旨在实现建模可靠性和计算效率之间的最佳权衡。在替代模型中，层次模型（HiMod）降阶为具有主导动力学方向的现象提供了一种有效方法。HiMod方法通过纳入横向动力学效应自然地获得一维模型。
HiMod降阶将沿主流方向的有限元近似与横向动力学的局部可调模态表示相结合。具体而言，我们关注逐点HiMod降阶策略，即在每个有限元节点上进行模态调整。我们通过采用一个在时间上不连续、在空间上连续且分层降阶的模型（c[M([公式：见正文])G()]-dG()近似），在非稳态情况下形式化逐点HiMod方法。模态分布和时空离散化的选择通过基于全局误差分析的自适应过程自动进行。该过程的最终结果是一个名为 的表格，它设定了时间划分，并针对每个时间间隔，给出相应的一维有限元网格以及相关的模态分布。
数值验证结果证实了所提出的自适应过程在准确性、对目标量和边界条件的敏感性以及计算节省方面的稳健性。最后，在地下水实验环境中的验证结果很有前景。
将HiMod降阶扩展到非稳态框架是迈向实际工程应用的关键一步。此外，验证阶段的结果证实HiMod近似是一种可行的方法。
